# A capital in Doge’s Palace in Venice

Doge’s Palace in Piazza San Marco is one of the most touristic attractions of Venice. The palace (XIVth c.) is very beaytiful and there is a hidden mathematical secret in one of the capitals of its columns. The capitals of the columns of the palace are dedicated to some biblical passages, quotidian Medieval scenes and… there is one capital dedicated to the Liberal Arts! So we can find here our famous three most representative figures of the Arithmetic, the Geometry and the Astronomy:

Pythagoras is counting money and next to his coins we can read the number 1399 and Euclid has a compass in one of his hands. The column is the first one next to the corner in front of the sea:

You must see this column in Venice!

**Location**: Piazza San Marco (map)

# The wonderful vault of a Royal Library

The King Philip II of Spain decided in 1550’s that he wanted to have a great library near his court in Madrid and he chose the new Monastery of San Lorenzo de El Escorial to place it in spite of other bigger villages. He didn’t want that the new library was a regular room inside a monastery so it had to be a very important place. Therefore, the library was placed on the second floor of the monastery just above his royal chambers but never above the basilica. Between 1565 and 1576, the king bought almost 5.000 books and manuscripts and the library became one of the most important libraries in all Europe.

The mathematician and architect Juan de Herrera (1530-1597) designed a large room (54 m. long x 9 m. wide x 10 m. high) with big windows in both sides under a great barrel vault. This vault had to be decorated by an important painter and Philip II decided that Peregrino Tibaldi (1527–1596) had to be the right artist to do the work. Philip II was advised by Juan de Herrera and other humanists and he decided that the main subject of the paintings of the vault had to be the Liberal Arts. Furthermore, the seven arts would be together with the Philosophy and the Theology on both ends of the room. The Philosophy represented the compendium of the Human knowledge and she is accompanied by Aristotle, Plato, Seneca and Socrates:

The Theology is on the side next to the convent and she represented the Divine knowledge. Therefore the vault represented the way from the Human Philosophy to the Divine knowledge through the seven Liberal Arts: the Arithmetic, the Geometry, the Astronomy (Astrology), the Music, the Rhetoric, the Grammar and the Dialectic. We can see a mathematical detail on the fresco below the Philosophy: it represents the School of Athens and there is a discussion between the Academics leaded by Socrates and the Stoics leaded by Zeno of Elea.

The scholars aren’t listening to the speakers because each of them is “playing” with something different. We can see at the lower left corner a man measuring something with a compass and two books, a sphere and an armilar sphere, a dodecahedron and a compass in the middle of the picture:

Going from the Philosophy to the Theology, we arrive at the Arithmetic after admiring the Grammar, the Rhetoric and the Dialectic. The Arithmetic is a woman turned to a table with simple mathematical operations rounded by muscled young men with tablets with arithmetical operations ans counting with their fingers:

There is also a representation of the Queen of Saba talking with King Solomon According to the Book of the Kings (I,10,1), the Queen of Saba went to meet Solomon to ask some enigmas to him so we can see a ruler, a balance and a tablet with some numbers written on it. In the red tablecloth we can read “Everything has number, weight and measure” in Hebrew:

The other panel next to the Arithmetic represents the school of the Gymnosophists who lived near the Nile and thought their philosophical theories from the numerical computations. In the middle of the picture we can see one of the gymnosophist with a compass looking at a triangle with the word “Anima” and the arithmetic progression 1, 2, 3 and 4 and the geometric 1, 3, 9 and 27 written on it. The other gymnosophists are computing with numbers written on the sand:

Finally, at both sides of the Arithmetic on the roof we find four people related with this subject: Archytas of Tarentum (c.428–c.347 BC) and Boethius (c.480-c.525) in one side and the Platonic Xenocrates (c.396/5 – 314/3 BC) and Jordan in the other. They are writing numbers in their tablets.

There is the Music after the Arithmetic and we find the Geometry after it:

She has a compass in one of her hands and the young men around her have different geometrical instruments. The two scenes which are on the corresponding walls next to her are dedicated to some Egyptian monks drawing geometrical figures on the sand…

and Archimedes’ death:

Notice that Archimedes is drawing the demonstration of the Theorem of Pythagoras made by Euclid!

Finally, the four chosen figures are the Astronomer Aristarchus of Samos (IIIrd c. BC) and the Persian astrologer Abd del Aziz also known as Alcabitius (Xth century) in one side and Archimedes (c.287-212 BC) and Regiomontanus (1436-1476) in the other. Aristarchus is measuring angles and has a dodecahedron at his feet, Alcabitius has a carpenter’s square, Archimedes has a compass and a sphere to measure the Earth and Regiomontanus is pointing at a dodecahedron.

The last Liberal Art is the Astrology. She is backed on a terrestrial globe and her eyes are looking at the sky. She has a compass in one of her hands and the little boys around her have an armilar sphere and some astronomical books:

In one of the two panels on the walls we can see Dionysius the Areopagite observing a solar eclipse the day of Jesuschrist’s death in Athens (Luke, 23,45) We can notice a quadrant and an astrolabe in the hands of the amazed men!

The other fresco represents King Ezekiel resting in bed and looking how time is delayed 15 years by God because of the repentance of his sins:

The four famous men are Euclid, Ptolemy, Alfonso X and Johannes of Sacrobosco. Euclid is represented here meaning the relationship between Astrology and Geometry. He has drawn three geometrical schemes. One is a triangle and a square inscribed in a circle and another square. Another scheme seems to be two overlaid squares partially hidden by Euclid’s name. In the middle of both pictures there is a man measuring the stars. Johannes of Sacrobosco has a quadrant in his right hand.

King Alfonso X of Castile (XIIIth. c) is the author of the *Libros del Saber de Astronomía* (“Books of the Astronomical knowledge”) and on the tablet which he has in his hands we notice a compass and the Ursa Maior (the compass is anachronistic!). His left hand has an open book with a horoscope

So you can see that this wonderful vault is an open mathematical book designed by Tibaldi and Juan de Herrera. I’ve been twice in the library and now I am waiting for the next time that I could enjoy this artistic part of the monastery of San Lorenzo de El Escorial.

**Location**: San Lorenzo de El Escorial (map)

# Del Castillo’s Arithmetic

The work exhibited in room 88 belongs to a set of six pictures painted for tapestries to decorate the cabinet of the Princess of Asturias at the Palacio Real del Pardo. The small children represent several allegories: Painting, Architecture, Fame, Music, Astronomy and Arithmetic. The composition was painted at the end of the Baroc style in the eighteenth century.

**Location**: Museo del Prado (map)

# The Liberal Arts in El Prado

In room 56B of the Museo del Prado we can admire Giovanni dal Ponte’s *Seven Liberal Arts*. There are some masterpieces in the same room 56B as Fra Angelico’s *Annuntiation *and therefore people don’t use to stop in front of this mathematical panel. In the web of the museum we can read:

This decoration of the front of a chest depicts the seven Liberal Arts, accompanied by an equal number of figures that represent the most relevant personages in each discipline. All are being crowned with laurel wreaths by small angels.

Astronomy presides over the composition, carrying the heavenly sphere, with Ptolemy (first and second centuries A.D.) sitting at his feet and reading one of the thirteen volumes in which he surveyed the history of Greek astronomy. To the right, Geometry holds an angle iron and a compass, walking hand-in-hand with Euclid (fourth and third centuries B.C.). He is followed by Arithmetic, who carries a counting tablet and is accompanied by Pythagoras (sixth century B.C.). At the right end of the composition, Music bears an organ, followed by its inventor, Tubalcain. To the left of Astronomy, Rhetoric carries a scroll and is accompanied by Cicero (first century B.C.), who carries one of his texts. Then comes Dialectics, who carries an olive branch as a symbol of agreement among the Arts, and a scorpion, whose pincers represent the opposing positions of dialectical thought. He is accompanied by Aristotle. At the left end of the composition is Grammar, with its disciplines, preceeded by two children and accompanied by Donatus (fourth century A.D.) or Priscian (fifth and sixth centuries A.D.).

This work exemplifies the coexistence in the arts of that period between the late Gothic heritage —visible in the use of gold and lineal calligraphy— and the new Renaissance style, which is clear in the solid and monumental definition of the figures, recalling works by Masaccio (1401-1428)

Our Ptolemy, Pythagoras and Euclid are the guest stars again and we have here a mathematical reason to visit room 56B. For example, Euclid is following the Geometry who is wearing a ruler and a compass:

After Euclid, Pythagoras is following the Arithmetic who holds a counting tablet:

Finally, Ptolemy is below the Astronomy:

**Location**: Museo del Prado (map)

# And finally… the Astronomy!

The third mathematical chapter of the De la Hire’s Liberal Arts is the Astronomy. We can see in the painting a lot of astronomical instruments and books!

Location: Musée des Beaux-Arts (map)

# De la Hire’s Geometry

De la Hire also painted the allegories of the Geometry and the Astronomy. The Geometry is a young woman with a paper in her left hand in which we can see some geometrical constructions as the famous Euclid’s demonstration of the theorem of Pythagoras:

In her left hand, she also holds a right angle edge and a compass. We can also see that there is a sphinx and an Egyptian background in the right which represents the Egyptian origin of the Geometry. Proclus stated that:

Since, then, we have to consider the beginnings of the arts and sciences with reference to the particular cycle [of the series postulated by Aristotle] through which the universe is at present passing, we say that, according to most accounts, geometry was first discovered in Egypt, having had its origin in the measurement of areas. For this was a necessity for the Egyptians owing to the rising of the Nile which effaced the proper boundaries of everybody’s lands.

Herodotus says that Ramses II distributed the land among the Egyptians in equal rectangular plots on which he levied an annual tax. When therefore the river swept away a portion of a plot and the owner applied for a corresponding reduction in the tax, surveyors had to sent down to certify what the reduction of the area had been.

The Geometry is next to a globe which nods to the science devoted o measuring the Earth (Geo + metry = Earth + measurement). There is also a snake representing the ancient goddess of the Earth.

**Location**: The Toledo Art Museum (map)

# De la Hire’s Arithmetic

Today is March 18, 2013 and Philippe de la Hire was born on March 18, 1640. Philippe de la Hire was a French mathematician who worked on astronomy and the conic sections. He provided an exposition of the properties of the conic sections and he applied the analytic geometry to some indeterminate problems about intersection of curves.

Philippe de la Hire’s father was the painter Laurent de la Hire (1606-1656) who got a lot of commissions from distinguished politicians, the Church and rich Parisian who wanted to have a portrait in their houses. But Laurent de la Hire had also time to paint the Allegories of the Liberal Arts and the *Arithmetic* can be enjoyed in the Walters Arts Museum of Baltimore.

The Arithmetic holds a book in which we can read the name of Pythagoras (c. 570 aC – c. 495 aC) and there is a paper on the book with the words “par” (“even”) and “impar” (“odd”) and an addition, difference and a multiplication. Can you imagine the young Philippe watching his father painting this picture? Maybe Philippe found his way to Mathematics in that moment!

**Location**: The Walters Art Museum (map)

# The Liberal Arts in the Campanile

Once again, the Liberal Arts are represented in a very famous monument. Now, I focus the attention in Giotto’s Campanile in Firenze, near the representation of Euclid:

The Geometry:

And the Astronomy:

**Location**: Giotto’s Campanile (map)

# The Spanish Chapel in Santa Maria Novella

The Spanish Chapel is one of the most wonderful chapels which can be enjoyed in Santa maria Novella. My students didn’t want to visit it but the teacher could convince most of them to enter the church and they weren’t disappointed.

The fresco entitled *The Triumph of St. Thomas Aquinas* was painted by Andrea di Bonaiuto (1365-1367) and it was dedicated to…

the great Dominican Doctor of the Church who, illuminated by the spirit of Wisdom, as described in the book lying open in his hands, and supported by the Theological and Cardinal Virtues and the study of the biblical writers of both the Old and New Testaments, defeats heresy, personified by Nestor, Arius and Averroes, and dominates the sciences. These are represented by fourteen allegorical female figures, alluding in part to the Sacred Sciences (left) and in part to the Liberal Arts (right). Each of these is accompanied by a historical personage, famous for having distinguished himself in that articular discipline.

The Theological and Cardinal Virtues are the Charity (over St. Thomas), the Faith and the Hope (at her respectively left and right sides), the Prudence (below the Faith), the Temperance (at the left side of the Prudence), The Justice (below the Hope) and the Fortitude (at her right). On the left of St. Thomas, there are (from left to right) the Biblical authors Job, David, St. Paul, St. Mark and St. Matthew, and on his right (from left to right), St. John the Evangelist, St. Luke, Moses, Isaiah and Solomon. Below St. Thomas, we find Nestor, Arius and Averroes:

The fourteen allegorical women and the corresponding eminent men are (from left to right): the Civil Law with Justinian, the Canonical Law with Clement V, the Philosophy with Aristotle, the Holy Scriptures with St. Jerome, the Theology with St. John of Damascus, the Contemplation with St. Dionysius the Areopagite, the Preaching with St. Augustine, the Arithmetic with Pythagoras, the Geometry with Euclid, the Astronomy with Ptolemy, the Music with Tubalcain, the Dialectics with Pietro Ispano (?), the Rhetoric with Cicero and the Grammar with Priscian (?):

Finally, here you are my privileged students who enjoyed the wonderful Spanish Chapel:

**Location**: Santa Maria Novella (map)

# A Parisian Arithmetic

This medieval Flemish tapestry is entitled *L’Arithmétique *and it is part of an anonymous series on the theme of the seven liberal arts. The composition is organized around a young woman standing behind a table. This woman is teaching some traders and bankers how to manage with the Arabic figures. The inscription says:

Monstrat ars numeri que virtus possit habere. / Explico per que sit proportio rerum.

I should look for the Geometry and the Astronomy to complete this allegorical series, don’t I?

**Location**: Musée of Cluny at Paris (map)